In magnetic resonance imaging, the object of interest is positioned in an image region through which passes a strong, substantially uniform, longitudinal magnetic field. Magnetic dipoles of nuclei disposed within the imaging region would then tend to align with the strong magnetic field. Resonance excitation frequency pulses are applied to the image region to place magnetization from the magnetic dipoles into the traverse plane where they continue to precess about the strong magnetic field. As the magnetic dipoles precess, corresponding radio frequency resonance signals are generated.
Various techniques, which are well known in the art, may be implemented to excite resonance and spatially encode the resonance signals. These techniques are accomplished by applying gradient magnetic fields in one, two and/or three directions. In one technique, orthogonal magnetic field gradients are applied to encode spatial position into a selected slice or planar region of interest. The magnetic field gradients are encoded along one axis giving a frequency which varies linearly in accordance with position and along an orthogonal axis or axes giving a phase that varies linearly in accordance with position. Accordingly, the spatial encoding is carried by the frequency and phase of the resonance signal. The resonance signal is collected for each of a plurality of phase encodings and transformed from a frequency and phase domain to an image or spatial domain. In the image domain, each transformed resonance signal (after the transforms along the phase encoding directions) produces a view representing a combination of the density and relaxation times (herein the dependence of the image on density and relaxation times are conjunctively termed spin density) of resonating nuclei in each pixel or voxel of the image region. The accuracy with which an image can be reconstructed following this technique generally is limited by the number of phase and frequency encoding steps, i.e., the number of collected views or sampled points.
In general, regardless of the particular technique applied to acquire the data, the acquisition of the data takes time. Usually a first spatial direction, referred to as the read direction, is collected very quickly. The second spatial direction, referred to as the phase encoding direction, often takes a much longer time to acquire. Still additional time is needed should data be phase encoded in a third or slice select direction. In any case, due to time considerations, it is not always possible to collect a sufficient number of data points to acquire a desired resolution applying known techniques for mapping the data in the frequency and phase domain to the image or spatial domain. Also, signal-to-noise is often poor when high resolution is desired.
Heretofore, standard Fourier transform imaging techniques have been used to reconstruct the discretely sampled data to produce an image. The usual Fourier transform technique suffers seriously from truncated data because of ringing, i.e., the Gibbs phenomena. Similar difficulties occur with the generalized transform when the pixel sizes are not chosen correctly or when a continuous object is imaged. In general, the conventional Fourier analysis with a limited number of sampled points is subject to limited resolution and poor signal-to-noise, and these drawbacks are rendered more severe in the case of objects with sharp edges.